Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 22.5s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a - 0.5\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (log t) (- a 0.5))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + (log(t) * (a - 0.5));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + (log(t) * (a - 0.5d0))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + (Math.log(t) * (a - 0.5));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + (math.log(t) * (a - 0.5))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(log(t) * Float64(a - 0.5)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + (log(t) * (a - 0.5));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Final simplification 99.6%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a - 0.5\right) \]

Alternative 2: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-243}:\\ \;\;\;\;\left(\log y + \log \left(\frac{z}{\sqrt{t}}\right)\right) - t\\ \mathbf{elif}\;a \leq 0.49:\\ \;\;\;\;\left(\log z + \log \left(\frac{y}{\sqrt{t}}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* a (log t)) t)))
   (if (<= a -3.2e-9)
     t_1
     (if (<= a 1.06e-243)
       (- (+ (log y) (log (/ z (sqrt t)))) t)
       (if (<= a 0.49) (- (+ (log z) (log (/ y (sqrt t)))) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * log(t)) - t;
	double tmp;
	if (a <= -3.2e-9) {
		tmp = t_1;
	} else if (a <= 1.06e-243) {
		tmp = (log(y) + log((z / sqrt(t)))) - t;
	} else if (a <= 0.49) {
		tmp = (log(z) + log((y / sqrt(t)))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * log(t)) - t
    if (a <= (-3.2d-9)) then
        tmp = t_1
    else if (a <= 1.06d-243) then
        tmp = (log(y) + log((z / sqrt(t)))) - t
    else if (a <= 0.49d0) then
        tmp = (log(z) + log((y / sqrt(t)))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -3.2e-9) {
		tmp = t_1;
	} else if (a <= 1.06e-243) {
		tmp = (Math.log(y) + Math.log((z / Math.sqrt(t)))) - t;
	} else if (a <= 0.49) {
		tmp = (Math.log(z) + Math.log((y / Math.sqrt(t)))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a * math.log(t)) - t
	tmp = 0
	if a <= -3.2e-9:
		tmp = t_1
	elif a <= 1.06e-243:
		tmp = (math.log(y) + math.log((z / math.sqrt(t)))) - t
	elif a <= 0.49:
		tmp = (math.log(z) + math.log((y / math.sqrt(t)))) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -3.2e-9)
		tmp = t_1;
	elseif (a <= 1.06e-243)
		tmp = Float64(Float64(log(y) + log(Float64(z / sqrt(t)))) - t);
	elseif (a <= 0.49)
		tmp = Float64(Float64(log(z) + log(Float64(y / sqrt(t)))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -3.2e-9)
		tmp = t_1;
	elseif (a <= 1.06e-243)
		tmp = (log(y) + log((z / sqrt(t)))) - t;
	elseif (a <= 0.49)
		tmp = (log(z) + log((y / sqrt(t)))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -3.2e-9], t$95$1, If[LessEqual[a, 1.06e-243], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.49], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.20000000000000012e-9 or 0.48999999999999999 < a

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.8%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.8%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.8%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 81.0%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 57.6%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   7. Taylor expanded in a around inf 97.1%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   8. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   9. Simplified 97.1%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]

   if -3.20000000000000012e-9 < a < 1.06e-243

   1. Initial program 99.5%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.5%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      2. +-commutative 99.5%

         \[\leadsto \color{blue}{\left(\left(\log z - t\right) + \log \left(x + y\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      3. associate-+l+ 99.4%

         \[\leadsto \color{blue}{\left(\log z - t\right) + \left(\log \left(x + y\right) + \left(a - 0.5\right) \cdot \log t\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log t + \log \left(x + y\right)\right)} \]

      5. fma-def 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \log \left(x + y\right)\right)} \]

      6. sub-neg 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, \log t, \log \left(x + y\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(a + \color{blue}{-0.5}, \log t, \log \left(x + y\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) + \left(\log z - t\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \color{blue}{\left(\left(a + -0.5\right) \cdot \log t + \log \left(x + y\right)\right)} + \left(\log z - t\right) \]

      3. metadata-eval 99.4%

         \[\leadsto \left(\left(a + \color{blue}{\left(-0.5\right)}\right) \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      4. sub-neg 99.4%

         \[\leadsto \left(\color{blue}{\left(a - 0.5\right)} \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      5. associate-+r+ 99.5%

         \[\leadsto \color{blue}{\left(a - 0.5\right) \cdot \log t + \left(\log \left(x + y\right) + \left(\log z - t\right)\right)} \]

      6. associate--l+ 99.5%

         \[\leadsto \left(a - 0.5\right) \cdot \log t + \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right)} \]

      7. add-log-exp 36.9%

         \[\leadsto \color{blue}{\log \left(e^{\left(a - 0.5\right) \cdot \log t + \left(\left(\log \left(x + y\right) + \log z\right) - t\right)}\right)} \]

      8. exp-sum 35.7%

         \[\leadsto \log \color{blue}{\left(e^{\left(a - 0.5\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right)} \]

      9. sub-neg 35.7%

         \[\leadsto \log \left(e^{\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      10. metadata-eval 35.7%

          \[\leadsto \log \left(e^{\left(a + \color{blue}{-0.5}\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      11. *-commutative 35.7%

          \[\leadsto \log \left(e^{\color{blue}{\log t \cdot \left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      12. exp-to-pow 35.7%

          \[\leadsto \log \left(\color{blue}{{t}^{\left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      13. associate--l+ 35.7%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot e^{\color{blue}{\log \left(x + y\right) + \left(\log z - t\right)}}\right) \]

      14. exp-sum 35.7%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot \color{blue}{\left(e^{\log \left(x + y\right)} \cdot e^{\log z - t}\right)}\right) \]

   5. Applied egg-rr 36.1%

      \[\leadsto \color{blue}{\log \left({t}^{\left(a + -0.5\right)} \cdot \left(\left(x + y\right) \cdot \frac{z}{e^{t}}\right)\right)} \]

   6. Taylor expanded in x around 0 23.8%

      \[\leadsto \color{blue}{\log \left(\frac{y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)}{e^{t}}\right)} \]
   7. Step-by-step derivation

      1. log-div 23.1%

         \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)\right) - \log \left(e^{t}\right)} \]

      2. exp-to-pow 23.1%

         \[\leadsto \log \left(y \cdot \left(z \cdot \color{blue}{{t}^{\left(a - 0.5\right)}}\right)\right) - \log \left(e^{t}\right) \]

      3. sub-neg 23.1%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\color{blue}{\left(a + \left(-0.5\right)\right)}}\right)\right) - \log \left(e^{t}\right) \]

      4. metadata-eval 23.1%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + \color{blue}{-0.5}\right)}\right)\right) - \log \left(e^{t}\right) \]

      5. rem-log-exp 41.8%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - \color{blue}{t} \]

   8. Simplified 41.8%

      \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t} \]

   9. Taylor expanded in a around 0 41.8%

      \[\leadsto \log \left(y \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} \cdot z\right)}\right) - t \]
   10. Step-by-step derivation

       1. *-commutative 41.8%

          \[\leadsto \log \color{blue}{\left(\left(\sqrt{\frac{1}{t}} \cdot z\right) \cdot y\right)} - t \]

       2. log-prod 55.5%

          \[\leadsto \color{blue}{\left(\log \left(\sqrt{\frac{1}{t}} \cdot z\right) + \log y\right)} - t \]

       3. *-commutative 55.5%

          \[\leadsto \left(\log \color{blue}{\left(z \cdot \sqrt{\frac{1}{t}}\right)} + \log y\right) - t \]

       4. sqrt-div 55.5%

          \[\leadsto \left(\log \left(z \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{t}}}\right) + \log y\right) - t \]

       5. metadata-eval 55.5%

          \[\leadsto \left(\log \left(z \cdot \frac{\color{blue}{1}}{\sqrt{t}}\right) + \log y\right) - t \]

       6. un-div-inv 55.5%

          \[\leadsto \left(\log \color{blue}{\left(\frac{z}{\sqrt{t}}\right)} + \log y\right) - t \]

   11. Applied egg-rr 55.5%

       \[\leadsto \color{blue}{\left(\log \left(\frac{z}{\sqrt{t}}\right) + \log y\right)} - t \]

   if 1.06e-243 < a < 0.48999999999999999

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.4%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      2. +-commutative 99.4%

         \[\leadsto \color{blue}{\left(\left(\log z - t\right) + \log \left(x + y\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      3. associate-+l+ 99.4%

         \[\leadsto \color{blue}{\left(\log z - t\right) + \left(\log \left(x + y\right) + \left(a - 0.5\right) \cdot \log t\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log t + \log \left(x + y\right)\right)} \]

      5. fma-def 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \log \left(x + y\right)\right)} \]

      6. sub-neg 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, \log t, \log \left(x + y\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(a + \color{blue}{-0.5}, \log t, \log \left(x + y\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) + \left(\log z - t\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \color{blue}{\left(\left(a + -0.5\right) \cdot \log t + \log \left(x + y\right)\right)} + \left(\log z - t\right) \]

      3. metadata-eval 99.4%

         \[\leadsto \left(\left(a + \color{blue}{\left(-0.5\right)}\right) \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      4. sub-neg 99.4%

         \[\leadsto \left(\color{blue}{\left(a - 0.5\right)} \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      5. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(a - 0.5\right) \cdot \log t + \left(\log \left(x + y\right) + \left(\log z - t\right)\right)} \]

      6. associate--l+ 99.4%

         \[\leadsto \left(a - 0.5\right) \cdot \log t + \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right)} \]

      7. add-log-exp 42.4%

         \[\leadsto \color{blue}{\log \left(e^{\left(a - 0.5\right) \cdot \log t + \left(\left(\log \left(x + y\right) + \log z\right) - t\right)}\right)} \]

      8. exp-sum 41.6%

         \[\leadsto \log \color{blue}{\left(e^{\left(a - 0.5\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right)} \]

      9. sub-neg 41.6%

         \[\leadsto \log \left(e^{\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      10. metadata-eval 41.6%

          \[\leadsto \log \left(e^{\left(a + \color{blue}{-0.5}\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      11. *-commutative 41.6%

          \[\leadsto \log \left(e^{\color{blue}{\log t \cdot \left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      12. exp-to-pow 41.6%

          \[\leadsto \log \left(\color{blue}{{t}^{\left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      13. associate--l+ 41.6%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot e^{\color{blue}{\log \left(x + y\right) + \left(\log z - t\right)}}\right) \]

      14. exp-sum 41.8%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot \color{blue}{\left(e^{\log \left(x + y\right)} \cdot e^{\log z - t}\right)}\right) \]

   5. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\log \left({t}^{\left(a + -0.5\right)} \cdot \left(\left(x + y\right) \cdot \frac{z}{e^{t}}\right)\right)} \]

   6. Taylor expanded in x around 0 21.7%

      \[\leadsto \color{blue}{\log \left(\frac{y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)}{e^{t}}\right)} \]
   7. Step-by-step derivation

      1. log-div 21.6%

         \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)\right) - \log \left(e^{t}\right)} \]

      2. exp-to-pow 21.7%

         \[\leadsto \log \left(y \cdot \left(z \cdot \color{blue}{{t}^{\left(a - 0.5\right)}}\right)\right) - \log \left(e^{t}\right) \]

      3. sub-neg 21.7%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\color{blue}{\left(a + \left(-0.5\right)\right)}}\right)\right) - \log \left(e^{t}\right) \]

      4. metadata-eval 21.7%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + \color{blue}{-0.5}\right)}\right)\right) - \log \left(e^{t}\right) \]

      5. rem-log-exp 38.7%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - \color{blue}{t} \]

   8. Simplified 38.7%

      \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t} \]

   9. Taylor expanded in a around 0 38.4%

      \[\leadsto \log \left(y \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} \cdot z\right)}\right) - t \]
   10. Step-by-step derivation

       1. associate-*r* 40.5%

          \[\leadsto \log \color{blue}{\left(\left(y \cdot \sqrt{\frac{1}{t}}\right) \cdot z\right)} - t \]

       2. log-prod 57.9%

          \[\leadsto \color{blue}{\left(\log \left(y \cdot \sqrt{\frac{1}{t}}\right) + \log z\right)} - t \]

       3. sqrt-div 57.9%

          \[\leadsto \left(\log \left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{t}}}\right) + \log z\right) - t \]

       4. metadata-eval 57.9%

          \[\leadsto \left(\log \left(y \cdot \frac{\color{blue}{1}}{\sqrt{t}}\right) + \log z\right) - t \]

       5. un-div-inv 57.9%

          \[\leadsto \left(\log \color{blue}{\left(\frac{y}{\sqrt{t}}\right)} + \log z\right) - t \]

   11. Applied egg-rr 57.9%

       \[\leadsto \color{blue}{\left(\log \left(\frac{y}{\sqrt{t}}\right) + \log z\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-243}:\\ \;\;\;\;\left(\log y + \log \left(\frac{z}{\sqrt{t}}\right)\right) - t\\ \mathbf{elif}\;a \leq 0.49:\\ \;\;\;\;\left(\log z + \log \left(\frac{y}{\sqrt{t}}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 3: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2300000000000 \lor \neg \left(a \leq 0.68\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(\frac{y}{\sqrt{t}}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2300000000000.0) (not (<= a 0.68)))
   (- (* a (log t)) t)
   (- (+ (log z) (log (/ y (sqrt t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2300000000000.0) || !(a <= 0.68)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + log((y / sqrt(t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2300000000000.0d0)) .or. (.not. (a <= 0.68d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + log((y / sqrt(t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2300000000000.0) || !(a <= 0.68)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + Math.log((y / Math.sqrt(t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2300000000000.0) or not (a <= 0.68):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + math.log((y / math.sqrt(t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2300000000000.0) || !(a <= 0.68))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + log(Float64(y / sqrt(t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2300000000000.0) || ~((a <= 0.68)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + log((y / sqrt(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2300000000000.0], N[Not[LessEqual[a, 0.68]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3e12 or 0.680000000000000049 < a

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.8%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.8%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.8%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 81.7%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 57.9%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   7. Taylor expanded in a around inf 99.1%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   8. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]

   if -2.3e12 < a < 0.680000000000000049

   1. Initial program 99.5%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.5%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      2. +-commutative 99.5%

         \[\leadsto \color{blue}{\left(\left(\log z - t\right) + \log \left(x + y\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      3. associate-+l+ 99.4%

         \[\leadsto \color{blue}{\left(\log z - t\right) + \left(\log \left(x + y\right) + \left(a - 0.5\right) \cdot \log t\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log t + \log \left(x + y\right)\right)} \]

      5. fma-def 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \log \left(x + y\right)\right)} \]

      6. sub-neg 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, \log t, \log \left(x + y\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(a + \color{blue}{-0.5}, \log t, \log \left(x + y\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) + \left(\log z - t\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \color{blue}{\left(\left(a + -0.5\right) \cdot \log t + \log \left(x + y\right)\right)} + \left(\log z - t\right) \]

      3. metadata-eval 99.4%

         \[\leadsto \left(\left(a + \color{blue}{\left(-0.5\right)}\right) \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      4. sub-neg 99.4%

         \[\leadsto \left(\color{blue}{\left(a - 0.5\right)} \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      5. associate-+r+ 99.5%

         \[\leadsto \color{blue}{\left(a - 0.5\right) \cdot \log t + \left(\log \left(x + y\right) + \left(\log z - t\right)\right)} \]

      6. associate--l+ 99.5%

         \[\leadsto \left(a - 0.5\right) \cdot \log t + \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right)} \]

      7. add-log-exp 38.1%

         \[\leadsto \color{blue}{\log \left(e^{\left(a - 0.5\right) \cdot \log t + \left(\left(\log \left(x + y\right) + \log z\right) - t\right)}\right)} \]

      8. exp-sum 37.2%

         \[\leadsto \log \color{blue}{\left(e^{\left(a - 0.5\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right)} \]

      9. sub-neg 37.2%

         \[\leadsto \log \left(e^{\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      10. metadata-eval 37.2%

          \[\leadsto \log \left(e^{\left(a + \color{blue}{-0.5}\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      11. *-commutative 37.2%

          \[\leadsto \log \left(e^{\color{blue}{\log t \cdot \left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      12. exp-to-pow 37.2%

          \[\leadsto \log \left(\color{blue}{{t}^{\left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      13. associate--l+ 37.1%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot e^{\color{blue}{\log \left(x + y\right) + \left(\log z - t\right)}}\right) \]

      14. exp-sum 37.2%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot \color{blue}{\left(e^{\log \left(x + y\right)} \cdot e^{\log z - t}\right)}\right) \]

   5. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{\log \left({t}^{\left(a + -0.5\right)} \cdot \left(\left(x + y\right) \cdot \frac{z}{e^{t}}\right)\right)} \]

   6. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{\log \left(\frac{y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)}{e^{t}}\right)} \]
   7. Step-by-step derivation

      1. log-div 21.7%

         \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)\right) - \log \left(e^{t}\right)} \]

      2. exp-to-pow 21.7%

         \[\leadsto \log \left(y \cdot \left(z \cdot \color{blue}{{t}^{\left(a - 0.5\right)}}\right)\right) - \log \left(e^{t}\right) \]

      3. sub-neg 21.7%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\color{blue}{\left(a + \left(-0.5\right)\right)}}\right)\right) - \log \left(e^{t}\right) \]

      4. metadata-eval 21.7%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + \color{blue}{-0.5}\right)}\right)\right) - \log \left(e^{t}\right) \]

      5. rem-log-exp 39.0%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - \color{blue}{t} \]

   8. Simplified 39.0%

      \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t} \]

   9. Taylor expanded in a around 0 40.3%

      \[\leadsto \log \left(y \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} \cdot z\right)}\right) - t \]
   10. Step-by-step derivation

       1. associate-*r* 41.2%

          \[\leadsto \log \color{blue}{\left(\left(y \cdot \sqrt{\frac{1}{t}}\right) \cdot z\right)} - t \]

       2. log-prod 54.1%

          \[\leadsto \color{blue}{\left(\log \left(y \cdot \sqrt{\frac{1}{t}}\right) + \log z\right)} - t \]

       3. sqrt-div 54.1%

          \[\leadsto \left(\log \left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{t}}}\right) + \log z\right) - t \]

       4. metadata-eval 54.1%

          \[\leadsto \left(\log \left(y \cdot \frac{\color{blue}{1}}{\sqrt{t}}\right) + \log z\right) - t \]

       5. un-div-inv 54.1%

          \[\leadsto \left(\log \color{blue}{\left(\frac{y}{\sqrt{t}}\right)} + \log z\right) - t \]

   11. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\left(\log \left(\frac{y}{\sqrt{t}}\right) + \log z\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2300000000000 \lor \neg \left(a \leq 0.68\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(\frac{y}{\sqrt{t}}\right)\right) - t\\ \end{array} \]

Alternative 4: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 235:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 235.0)
   (+ (+ (log z) (log y)) (* (log t) (- a 0.5)))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 235.0) {
		tmp = (log(z) + log(y)) + (log(t) * (a - 0.5));
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 235.0d0) then
        tmp = (log(z) + log(y)) + (log(t) * (a - 0.5d0))
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 235.0) {
		tmp = (Math.log(z) + Math.log(y)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 235.0:
		tmp = (math.log(z) + math.log(y)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 235.0)
		tmp = Float64(Float64(log(z) + log(y)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 235.0)
		tmp = (log(z) + log(y)) + (log(t) * (a - 0.5));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 235.0], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 235

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.3%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.3%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]

   4. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{\left(\log y + \log z\right) - \left(t + \log t \cdot \left(0.5 - a\right)\right)} \]

   5. Taylor expanded in t around 0 61.9%

      \[\leadsto \color{blue}{\left(\log y + \log z\right) - \log t \cdot \left(0.5 - a\right)} \]

   if 235 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.9%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.9%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.9%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.9%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.9%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 77.7%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 59.6%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   7. Taylor expanded in a around inf 99.6%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 235:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\log z + \log y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (+ (log z) (log y)) (- (* (log t) (- a 0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	return (log(z) + log(y)) + ((log(t) * (a - 0.5)) - t);
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (log(z) + log(y)) + ((log(t) * (a - 0.5d0)) - t)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return (Math.log(z) + Math.log(y)) + ((Math.log(t) * (a - 0.5)) - t);
}

Python

def code(x, y, z, t, a):
	return (math.log(z) + math.log(y)) + ((math.log(t) * (a - 0.5)) - t)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(log(z) + log(y)) + Float64(Float64(log(t) * Float64(a - 0.5)) - t))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (log(z) + log(y)) + ((log(t) * (a - 0.5)) - t);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Step-by-step derivation

   1. associate-+l- 99.6%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

   2. +-commutative 99.6%

      \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

   3. associate--l+ 99.6%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

   4. sub-neg 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

   5. +-commutative 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

   6. *-commutative 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

   7. distribute-rgt-neg-in 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

   8. fma-def 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

   9. sub-neg 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

   10. +-commutative 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

   11. distribute-neg-in 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

   12. metadata-eval 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

   13. metadata-eval 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

   14. unsub-neg 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]

4. Taylor expanded in x around 0 67.1%

   \[\leadsto \color{blue}{\left(\log y + \log z\right) - \left(t + \log t \cdot \left(0.5 - a\right)\right)} \]

5. Final simplification 67.1%

   \[\leadsto \left(\log z + \log y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right) \]

Alternative 6: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 195000000000:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 195000000000.0)
   (- (+ (log (* y z)) (* (log t) (- a 0.5))) t)
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 195000000000.0) {
		tmp = (log((y * z)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 195000000000.0d0) then
        tmp = (log((y * z)) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 195000000000.0) {
		tmp = (Math.log((y * z)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 195000000000.0:
		tmp = (math.log((y * z)) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 195000000000.0)
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 195000000000.0)
		tmp = (log((y * z)) + (log(t) * (a - 0.5))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 195000000000.0], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.95e11

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.3%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.3%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.3%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.3%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.3%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.3%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 80.3%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 47.6%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   if 1.95e11 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.9%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.9%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.9%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.9%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.9%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 77.5%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 60.0%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   7. Taylor expanded in a around inf 99.9%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   8. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   9. Simplified 99.9%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 195000000000:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 7: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-40} \lor \neg \left(a \leq 0.038\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.5e-40) (not (<= a 0.038)))
   (- (* a (log t)) t)
   (- (log (* y (/ z (sqrt t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.5e-40) || !(a <= 0.038)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = log((y * (z / sqrt(t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-5.5d-40)) .or. (.not. (a <= 0.038d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = log((y * (z / sqrt(t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.5e-40) || !(a <= 0.038)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = Math.log((y * (z / Math.sqrt(t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5.5e-40) or not (a <= 0.038):
		tmp = (a * math.log(t)) - t
	else:
		tmp = math.log((y * (z / math.sqrt(t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.5e-40) || !(a <= 0.038))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(log(Float64(y * Float64(z / sqrt(t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.5e-40) || ~((a <= 0.038)))
		tmp = (a * log(t)) - t;
	else
		tmp = log((y * (z / sqrt(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.5e-40], N[Not[LessEqual[a, 0.038]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * N[(z / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.50000000000000002e-40 or 0.0379999999999999991 < a

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.8%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.8%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.8%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 80.7%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 57.0%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   7. Taylor expanded in a around inf 95.7%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   8. Step-by-step derivation

      1. *-commutative 95.7%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   9. Simplified 95.7%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]

   if -5.50000000000000002e-40 < a < 0.0379999999999999991

   1. Initial program 99.5%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.4%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      2. +-commutative 99.4%

         \[\leadsto \color{blue}{\left(\left(\log z - t\right) + \log \left(x + y\right)\right)} + \left(a - 0.5\right) \cdot \log t \]

      3. associate-+l+ 99.4%

         \[\leadsto \color{blue}{\left(\log z - t\right) + \left(\log \left(x + y\right) + \left(a - 0.5\right) \cdot \log t\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log t + \log \left(x + y\right)\right)} \]

      5. fma-def 99.4%

         \[\leadsto \left(\log z - t\right) + \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \log \left(x + y\right)\right)} \]

      6. sub-neg 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, \log t, \log \left(x + y\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \left(\log z - t\right) + \mathsf{fma}\left(a + \color{blue}{-0.5}, \log t, \log \left(x + y\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) + \left(\log z - t\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \color{blue}{\left(\left(a + -0.5\right) \cdot \log t + \log \left(x + y\right)\right)} + \left(\log z - t\right) \]

      3. metadata-eval 99.4%

         \[\leadsto \left(\left(a + \color{blue}{\left(-0.5\right)}\right) \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      4. sub-neg 99.4%

         \[\leadsto \left(\color{blue}{\left(a - 0.5\right)} \cdot \log t + \log \left(x + y\right)\right) + \left(\log z - t\right) \]

      5. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(a - 0.5\right) \cdot \log t + \left(\log \left(x + y\right) + \left(\log z - t\right)\right)} \]

      6. associate--l+ 99.5%

         \[\leadsto \left(a - 0.5\right) \cdot \log t + \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right)} \]

      7. add-log-exp 39.8%

         \[\leadsto \color{blue}{\log \left(e^{\left(a - 0.5\right) \cdot \log t + \left(\left(\log \left(x + y\right) + \log z\right) - t\right)}\right)} \]

      8. exp-sum 38.7%

         \[\leadsto \log \color{blue}{\left(e^{\left(a - 0.5\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right)} \]

      9. sub-neg 38.7%

         \[\leadsto \log \left(e^{\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      10. metadata-eval 38.7%

          \[\leadsto \log \left(e^{\left(a + \color{blue}{-0.5}\right) \cdot \log t} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      11. *-commutative 38.7%

          \[\leadsto \log \left(e^{\color{blue}{\log t \cdot \left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      12. exp-to-pow 38.7%

          \[\leadsto \log \left(\color{blue}{{t}^{\left(a + -0.5\right)}} \cdot e^{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      13. associate--l+ 38.7%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot e^{\color{blue}{\log \left(x + y\right) + \left(\log z - t\right)}}\right) \]

      14. exp-sum 38.7%

          \[\leadsto \log \left({t}^{\left(a + -0.5\right)} \cdot \color{blue}{\left(e^{\log \left(x + y\right)} \cdot e^{\log z - t}\right)}\right) \]

   5. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{\log \left({t}^{\left(a + -0.5\right)} \cdot \left(\left(x + y\right) \cdot \frac{z}{e^{t}}\right)\right)} \]

   6. Taylor expanded in x around 0 23.5%

      \[\leadsto \color{blue}{\log \left(\frac{y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)}{e^{t}}\right)} \]
   7. Step-by-step derivation

      1. log-div 23.0%

         \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot e^{\log t \cdot \left(a - 0.5\right)}\right)\right) - \log \left(e^{t}\right)} \]

      2. exp-to-pow 23.1%

         \[\leadsto \log \left(y \cdot \left(z \cdot \color{blue}{{t}^{\left(a - 0.5\right)}}\right)\right) - \log \left(e^{t}\right) \]

      3. sub-neg 23.1%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\color{blue}{\left(a + \left(-0.5\right)\right)}}\right)\right) - \log \left(e^{t}\right) \]

      4. metadata-eval 23.1%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + \color{blue}{-0.5}\right)}\right)\right) - \log \left(e^{t}\right) \]

      5. rem-log-exp 41.5%

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - \color{blue}{t} \]

   8. Simplified 41.5%

      \[\leadsto \color{blue}{\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t} \]

   9. Taylor expanded in a around 0 41.4%

      \[\leadsto \log \left(y \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} \cdot z\right)}\right) - t \]
   10. Step-by-step derivation

       1. expm1-log1p-u 41.4%

          \[\leadsto \log \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{t}} \cdot z\right)\right)}\right) - t \]

       2. expm1-udef 19.4%

          \[\leadsto \log \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{t}} \cdot z\right)} - 1\right)}\right) - t \]

       3. *-commutative 19.4%

          \[\leadsto \log \left(y \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{z \cdot \sqrt{\frac{1}{t}}}\right)} - 1\right)\right) - t \]

       4. sqrt-div 19.4%

          \[\leadsto \log \left(y \cdot \left(e^{\mathsf{log1p}\left(z \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{t}}}\right)} - 1\right)\right) - t \]

       5. metadata-eval 19.4%

          \[\leadsto \log \left(y \cdot \left(e^{\mathsf{log1p}\left(z \cdot \frac{\color{blue}{1}}{\sqrt{t}}\right)} - 1\right)\right) - t \]

       6. un-div-inv 19.4%

          \[\leadsto \log \left(y \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{z}{\sqrt{t}}}\right)} - 1\right)\right) - t \]

   11. Applied egg-rr 19.4%

       \[\leadsto \log \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{z}{\sqrt{t}}\right)} - 1\right)}\right) - t \]
   12. Step-by-step derivation

       1. expm1-def 41.4%

          \[\leadsto \log \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{\sqrt{t}}\right)\right)}\right) - t \]

       2. expm1-log1p 41.4%

          \[\leadsto \log \left(y \cdot \color{blue}{\frac{z}{\sqrt{t}}}\right) - t \]

   13. Simplified 41.4%

       \[\leadsto \log \left(y \cdot \color{blue}{\frac{z}{\sqrt{t}}}\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-40} \lor \neg \left(a \leq 0.038\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right) - t\\ \end{array} \]

Alternative 8: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.112:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.112) (+ (log (* y z)) (* (log t) (- a 0.5))) (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.112) {
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.112d0) then
        tmp = log((y * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.112) {
		tmp = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.112:
		tmp = math.log((y * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.112)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.112)
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.112], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.112000000000000002

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.3%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.3%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]

   4. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{\left(\log y + \log z\right) - \left(t + \log t \cdot \left(0.5 - a\right)\right)} \]

   5. Taylor expanded in t around 0 61.9%

      \[\leadsto \color{blue}{\left(\log y + \log z\right) - \log t \cdot \left(0.5 - a\right)} \]
   6. Step-by-step derivation

      1. log-prod 47.9%

         \[\leadsto \color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right) \]

      2. *-commutative 47.9%

         \[\leadsto \log \color{blue}{\left(z \cdot y\right)} - \log t \cdot \left(0.5 - a\right) \]

   7. Simplified 47.9%

      \[\leadsto \color{blue}{\log \left(z \cdot y\right) - \log t \cdot \left(0.5 - a\right)} \]

   if 0.112000000000000002 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.9%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.9%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.9%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.9%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.9%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 77.7%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 59.6%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   7. Taylor expanded in a around inf 99.6%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.112:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 9: 69.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2300000000000 \lor \neg \left(a \leq 2.6\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2300000000000.0) (not (<= a 2.6)))
   (- (* a (log t)) t)
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2300000000000.0) || !(a <= 2.6)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2300000000000.0d0)) .or. (.not. (a <= 2.6d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2300000000000.0) || !(a <= 2.6)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2300000000000.0) or not (a <= 2.6):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2300000000000.0) || !(a <= 2.6))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2300000000000.0) || ~((a <= 2.6)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2300000000000.0], N[Not[LessEqual[a, 2.6]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3e12 or 2.60000000000000009 < a

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.8%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.8%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.8%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r- 99.8%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

      3. fma-udef 99.8%

         \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

      4. associate--r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

      5. +-commutative 99.8%

         \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

      6. sum-log 81.7%

         \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   5. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   6. Taylor expanded in x around 0 57.9%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   7. Taylor expanded in a around inf 99.1%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   8. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]

   if -2.3e12 < a < 2.60000000000000009

   1. Initial program 99.5%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.5%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.4%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.4%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.4%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.4%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.4%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.4%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.4%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.4%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.4%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.4%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.4%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.4%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]

   4. Taylor expanded in x around 0 62.5%

      \[\leadsto \color{blue}{\left(\log y + \log z\right) - \left(t + \log t \cdot \left(0.5 - a\right)\right)} \]

   5. Taylor expanded in t around inf 38.2%

      \[\leadsto \left(\log y + \log z\right) - \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2300000000000 \lor \neg \left(a \leq 2.6\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]

Alternative 10: 61.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{+30}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 5.7e+30) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.7e+30) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 5.7d+30) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.7e+30) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 5.7e+30:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 5.7e+30)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 5.7e+30)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 5.7e+30], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 5.7000000000000002e30

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.4%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.4%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.3%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.3%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.3%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]

   4. Taylor expanded in a around inf 46.2%

      \[\leadsto \color{blue}{a \cdot \log t} \]
   5. Step-by-step derivation

      1. *-commutative 46.2%

         \[\leadsto \color{blue}{\log t \cdot a} \]

   6. Simplified 46.2%

      \[\leadsto \color{blue}{\log t \cdot a} \]

   if 5.7000000000000002e30 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

      3. associate--l+ 99.9%

         \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

      4. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

      5. +-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

      6. *-commutative 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

      7. distribute-rgt-neg-in 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

      8. fma-def 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

      9. sub-neg 99.9%

         \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

      10. +-commutative 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

      11. distribute-neg-in 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

      12. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

      14. unsub-neg 99.9%

          \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]

   4. Taylor expanded in t around inf 81.8%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   5. Step-by-step derivation

      1. neg-mul-1 81.8%

         \[\leadsto \color{blue}{-t} \]

   6. Simplified 81.8%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{+30}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 74.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- (* a (log t)) t))

C

double code(double x, double y, double z, double t, double a) {
	return (a * log(t)) - t;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (a * log(t)) - t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return (a * Math.log(t)) - t;
}

Python

def code(x, y, z, t, a):
	return (a * math.log(t)) - t

Julia

function code(x, y, z, t, a)
	return Float64(Float64(a * log(t)) - t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (a * log(t)) - t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Step-by-step derivation

   1. associate-+l- 99.6%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

   2. +-commutative 99.6%

      \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

   3. associate--l+ 99.6%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

   4. sub-neg 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

   5. +-commutative 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

   6. *-commutative 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

   7. distribute-rgt-neg-in 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

   8. fma-def 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

   9. sub-neg 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

   10. +-commutative 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

   11. distribute-neg-in 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

   12. metadata-eval 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

   13. metadata-eval 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

   14. unsub-neg 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
4. Step-by-step derivation

   1. associate-+r- 99.6%

      \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

   2. +-commutative 99.6%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \mathsf{fma}\left(\log t, 0.5 - a, t\right) \]

   3. fma-udef 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\log t \cdot \left(0.5 - a\right) + t\right)} \]

   4. associate--r+ 99.6%

      \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

   5. +-commutative 99.6%

      \[\leadsto \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

   6. sum-log 79.0%

      \[\leadsto \left(\color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

5. Applied egg-rr 79.0%

   \[\leadsto \color{blue}{\left(\log \left(z \cdot \left(x + y\right)\right) - \log t \cdot \left(0.5 - a\right)\right) - t} \]

6. Taylor expanded in x around 0 53.6%

   \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - \log t \cdot \left(0.5 - a\right)\right) - t \]

7. Taylor expanded in a around inf 70.8%

   \[\leadsto \color{blue}{a \cdot \log t} - t \]
8. Step-by-step derivation

   1. *-commutative 70.8%

      \[\leadsto \color{blue}{\log t \cdot a} - t \]

9. Simplified 70.8%

   \[\leadsto \color{blue}{\log t \cdot a} - t \]

10. Final simplification 70.8%

    \[\leadsto a \cdot \log t - t \]

Alternative 12: 37.4% accurate, 156.5× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- t))

C

double code(double x, double y, double z, double t, double a) {
	return -t;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -t;
}

Python

def code(x, y, z, t, a):
	return -t

Julia

function code(x, y, z, t, a)
	return Float64(-t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t;
end

Wolfram

code[x_, y_, z_, t_, a_] := (-t)

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Step-by-step derivation

   1. associate-+l- 99.6%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]

   2. +-commutative 99.6%

      \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - \left(t - \left(a - 0.5\right) \cdot \log t\right) \]

   3. associate--l+ 99.6%

      \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]

   4. sub-neg 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]

   5. +-commutative 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]

   6. *-commutative 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]

   7. distribute-rgt-neg-in 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

   8. fma-def 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

   9. sub-neg 99.6%

      \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

   10. +-commutative 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

   11. distribute-neg-in 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

   12. metadata-eval 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

   13. metadata-eval 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

   14. unsub-neg 99.6%

       \[\leadsto \log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\log z + \left(\log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]

4. Taylor expanded in t around inf 39.1%

   \[\leadsto \color{blue}{-1 \cdot t} \]
5. Step-by-step derivation

   1. neg-mul-1 39.1%

      \[\leadsto \color{blue}{-t} \]

6. Simplified 39.1%

   \[\leadsto \color{blue}{-t} \]

7. Final simplification 39.1%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	return log((x + y)) + ((log(z) - t) + ((a - 0.5) * log(t)));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = log((x + y)) + ((log(z) - t) + ((a - 0.5d0) * log(t)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return Math.log((x + y)) + ((Math.log(z) - t) + ((a - 0.5) * Math.log(t)));
}

Python

def code(x, y, z, t, a):
	return math.log((x + y)) + ((math.log(z) - t) + ((a - 0.5) * math.log(t)))

Julia

function code(x, y, z, t, a)
	return Float64(log(Float64(x + y)) + Float64(Float64(log(z) - t) + Float64(Float64(a - 0.5) * log(t))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = log((x + y)) + ((log(z) - t) + ((a - 0.5) * log(t)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023310 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))